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G = C52⋊D4order 416 = 25·13

3rd semidirect product of C52 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C523D4, Dic131D4, C23.10D26, (C2×D52)⋊9C2, (D4×C26)⋊4C2, (C2×D4)⋊6D13, C41(C13⋊D4), C132(C41D4), (C2×C4).52D26, C26.52(C2×D4), C2.28(D4×D13), (C4×Dic13)⋊6C2, (C2×C26).55C23, (C2×C52).35C22, (C22×C26).22C22, C22.62(C22×D13), (C2×Dic13).42C22, (C22×D13).12C22, (C2×C13⋊D4)⋊7C2, C2.16(C2×C13⋊D4), SmallGroup(416,161)

Series: Derived Chief Lower central Upper central

C1C2×C26 — C52⋊D4
C1C13C26C2×C26C22×D13C2×D52 — C52⋊D4
C13C2×C26 — C52⋊D4
C1C22C2×D4

Generators and relations for C52⋊D4
 G = < a,b,c | a52=b4=c2=1, bab-1=a25, cac=a-1, cbc=b-1 >

Subgroups: 840 in 108 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×12], C2×C4, C2×C4 [×2], D4 [×12], C23 [×2], C23 [×2], C13, C42, C2×D4, C2×D4 [×5], D13 [×2], C26, C26 [×2], C26 [×2], C41D4, Dic13 [×4], C52 [×2], D26 [×6], C2×C26, C2×C26 [×6], D52 [×2], C2×Dic13 [×2], C13⋊D4 [×8], C2×C52, D4×C13 [×2], C22×D13 [×2], C22×C26 [×2], C4×Dic13, C2×D52, C2×C13⋊D4 [×4], D4×C26, C52⋊D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], D13, C41D4, D26 [×3], C13⋊D4 [×2], C22×D13, D4×D13 [×2], C2×C13⋊D4, C52⋊D4

Smallest permutation representation of C52⋊D4
On 208 points
Generators in S208
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52)(53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)
(1 69 156 172)(2 94 105 197)(3 67 106 170)(4 92 107 195)(5 65 108 168)(6 90 109 193)(7 63 110 166)(8 88 111 191)(9 61 112 164)(10 86 113 189)(11 59 114 162)(12 84 115 187)(13 57 116 160)(14 82 117 185)(15 55 118 158)(16 80 119 183)(17 53 120 208)(18 78 121 181)(19 103 122 206)(20 76 123 179)(21 101 124 204)(22 74 125 177)(23 99 126 202)(24 72 127 175)(25 97 128 200)(26 70 129 173)(27 95 130 198)(28 68 131 171)(29 93 132 196)(30 66 133 169)(31 91 134 194)(32 64 135 167)(33 89 136 192)(34 62 137 165)(35 87 138 190)(36 60 139 163)(37 85 140 188)(38 58 141 161)(39 83 142 186)(40 56 143 159)(41 81 144 184)(42 54 145 157)(43 79 146 182)(44 104 147 207)(45 77 148 180)(46 102 149 205)(47 75 150 178)(48 100 151 203)(49 73 152 176)(50 98 153 201)(51 71 154 174)(52 96 155 199)
(1 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 175)(54 174)(55 173)(56 172)(57 171)(58 170)(59 169)(60 168)(61 167)(62 166)(63 165)(64 164)(65 163)(66 162)(67 161)(68 160)(69 159)(70 158)(71 157)(72 208)(73 207)(74 206)(75 205)(76 204)(77 203)(78 202)(79 201)(80 200)(81 199)(82 198)(83 197)(84 196)(85 195)(86 194)(87 193)(88 192)(89 191)(90 190)(91 189)(92 188)(93 187)(94 186)(95 185)(96 184)(97 183)(98 182)(99 181)(100 180)(101 179)(102 178)(103 177)(104 176)(105 142)(106 141)(107 140)(108 139)(109 138)(110 137)(111 136)(112 135)(113 134)(114 133)(115 132)(116 131)(117 130)(118 129)(119 128)(120 127)(121 126)(122 125)(123 124)(143 156)(144 155)(145 154)(146 153)(147 152)(148 151)(149 150)

G:=sub<Sym(208)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,69,156,172)(2,94,105,197)(3,67,106,170)(4,92,107,195)(5,65,108,168)(6,90,109,193)(7,63,110,166)(8,88,111,191)(9,61,112,164)(10,86,113,189)(11,59,114,162)(12,84,115,187)(13,57,116,160)(14,82,117,185)(15,55,118,158)(16,80,119,183)(17,53,120,208)(18,78,121,181)(19,103,122,206)(20,76,123,179)(21,101,124,204)(22,74,125,177)(23,99,126,202)(24,72,127,175)(25,97,128,200)(26,70,129,173)(27,95,130,198)(28,68,131,171)(29,93,132,196)(30,66,133,169)(31,91,134,194)(32,64,135,167)(33,89,136,192)(34,62,137,165)(35,87,138,190)(36,60,139,163)(37,85,140,188)(38,58,141,161)(39,83,142,186)(40,56,143,159)(41,81,144,184)(42,54,145,157)(43,79,146,182)(44,104,147,207)(45,77,148,180)(46,102,149,205)(47,75,150,178)(48,100,151,203)(49,73,152,176)(50,98,153,201)(51,71,154,174)(52,96,155,199), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,175)(54,174)(55,173)(56,172)(57,171)(58,170)(59,169)(60,168)(61,167)(62,166)(63,165)(64,164)(65,163)(66,162)(67,161)(68,160)(69,159)(70,158)(71,157)(72,208)(73,207)(74,206)(75,205)(76,204)(77,203)(78,202)(79,201)(80,200)(81,199)(82,198)(83,197)(84,196)(85,195)(86,194)(87,193)(88,192)(89,191)(90,190)(91,189)(92,188)(93,187)(94,186)(95,185)(96,184)(97,183)(98,182)(99,181)(100,180)(101,179)(102,178)(103,177)(104,176)(105,142)(106,141)(107,140)(108,139)(109,138)(110,137)(111,136)(112,135)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)(119,128)(120,127)(121,126)(122,125)(123,124)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208), (1,69,156,172)(2,94,105,197)(3,67,106,170)(4,92,107,195)(5,65,108,168)(6,90,109,193)(7,63,110,166)(8,88,111,191)(9,61,112,164)(10,86,113,189)(11,59,114,162)(12,84,115,187)(13,57,116,160)(14,82,117,185)(15,55,118,158)(16,80,119,183)(17,53,120,208)(18,78,121,181)(19,103,122,206)(20,76,123,179)(21,101,124,204)(22,74,125,177)(23,99,126,202)(24,72,127,175)(25,97,128,200)(26,70,129,173)(27,95,130,198)(28,68,131,171)(29,93,132,196)(30,66,133,169)(31,91,134,194)(32,64,135,167)(33,89,136,192)(34,62,137,165)(35,87,138,190)(36,60,139,163)(37,85,140,188)(38,58,141,161)(39,83,142,186)(40,56,143,159)(41,81,144,184)(42,54,145,157)(43,79,146,182)(44,104,147,207)(45,77,148,180)(46,102,149,205)(47,75,150,178)(48,100,151,203)(49,73,152,176)(50,98,153,201)(51,71,154,174)(52,96,155,199), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,175)(54,174)(55,173)(56,172)(57,171)(58,170)(59,169)(60,168)(61,167)(62,166)(63,165)(64,164)(65,163)(66,162)(67,161)(68,160)(69,159)(70,158)(71,157)(72,208)(73,207)(74,206)(75,205)(76,204)(77,203)(78,202)(79,201)(80,200)(81,199)(82,198)(83,197)(84,196)(85,195)(86,194)(87,193)(88,192)(89,191)(90,190)(91,189)(92,188)(93,187)(94,186)(95,185)(96,184)(97,183)(98,182)(99,181)(100,180)(101,179)(102,178)(103,177)(104,176)(105,142)(106,141)(107,140)(108,139)(109,138)(110,137)(111,136)(112,135)(113,134)(114,133)(115,132)(116,131)(117,130)(118,129)(119,128)(120,127)(121,126)(122,125)(123,124)(143,156)(144,155)(145,154)(146,153)(147,152)(148,151)(149,150) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)], [(1,69,156,172),(2,94,105,197),(3,67,106,170),(4,92,107,195),(5,65,108,168),(6,90,109,193),(7,63,110,166),(8,88,111,191),(9,61,112,164),(10,86,113,189),(11,59,114,162),(12,84,115,187),(13,57,116,160),(14,82,117,185),(15,55,118,158),(16,80,119,183),(17,53,120,208),(18,78,121,181),(19,103,122,206),(20,76,123,179),(21,101,124,204),(22,74,125,177),(23,99,126,202),(24,72,127,175),(25,97,128,200),(26,70,129,173),(27,95,130,198),(28,68,131,171),(29,93,132,196),(30,66,133,169),(31,91,134,194),(32,64,135,167),(33,89,136,192),(34,62,137,165),(35,87,138,190),(36,60,139,163),(37,85,140,188),(38,58,141,161),(39,83,142,186),(40,56,143,159),(41,81,144,184),(42,54,145,157),(43,79,146,182),(44,104,147,207),(45,77,148,180),(46,102,149,205),(47,75,150,178),(48,100,151,203),(49,73,152,176),(50,98,153,201),(51,71,154,174),(52,96,155,199)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,175),(54,174),(55,173),(56,172),(57,171),(58,170),(59,169),(60,168),(61,167),(62,166),(63,165),(64,164),(65,163),(66,162),(67,161),(68,160),(69,159),(70,158),(71,157),(72,208),(73,207),(74,206),(75,205),(76,204),(77,203),(78,202),(79,201),(80,200),(81,199),(82,198),(83,197),(84,196),(85,195),(86,194),(87,193),(88,192),(89,191),(90,190),(91,189),(92,188),(93,187),(94,186),(95,185),(96,184),(97,183),(98,182),(99,181),(100,180),(101,179),(102,178),(103,177),(104,176),(105,142),(106,141),(107,140),(108,139),(109,138),(110,137),(111,136),(112,135),(113,134),(114,133),(115,132),(116,131),(117,130),(118,129),(119,128),(120,127),(121,126),(122,125),(123,124),(143,156),(144,155),(145,154),(146,153),(147,152),(148,151),(149,150)])

74 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F13A···13F26A···26R26S···26AP52A···52L
order1222222244444413···1326···2626···2652···52
size111144525222262626262···22···24···44···4

74 irreducible representations

dim111112222224
type+++++++++++
imageC1C2C2C2C2D4D4D13D26D26C13⋊D4D4×D13
kernelC52⋊D4C4×Dic13C2×D52C2×C13⋊D4D4×C26Dic13C52C2×D4C2×C4C23C4C2
# reps111414266122412

Matrix representation of C52⋊D4 in GL4(𝔽53) generated by

341200
151000
005013
00403
,
6900
434700
0010
0001
,
15200
473800
005013
00323
G:=sub<GL(4,GF(53))| [34,15,0,0,12,10,0,0,0,0,50,40,0,0,13,3],[6,43,0,0,9,47,0,0,0,0,1,0,0,0,0,1],[15,47,0,0,2,38,0,0,0,0,50,32,0,0,13,3] >;

C52⋊D4 in GAP, Magma, Sage, TeX

C_{52}\rtimes D_4
% in TeX

G:=Group("C52:D4");
// GroupNames label

G:=SmallGroup(416,161);
// by ID

G=gap.SmallGroup(416,161);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,217,103,218,188,13829]);
// Polycyclic

G:=Group<a,b,c|a^52=b^4=c^2=1,b*a*b^-1=a^25,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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